re: #727 itellu3times

The perfection of the number system does not depend on the imperfections of the real world. Plato knew that. Too bad your PhD did not.

I’m no expert on this, but a logician or number theorist could recite this stuff. The symbol for 1 is a perfect one - because we say it is! If we say we have one carrot, then that’s what we say. But then, we lie a lot, and are mistaken more often, it sometimes seems it’s almost luck when we get something right.

The number one plays its role in various algebraic or logical systems.

In Russell and Whitehead’s Principia Mathematica, I believe they spend a lot of time on establishing these things. What the status of their writing is today, I’m not sure. I think something like my arguments above, insofar as I got it at all right, are newer.

Principia Mathematica has been falsified by Kurt Godel, who proved that any logical system complex enpugh to permit self-referential statements must contain undecideable propositions, and therefore must be either incorrect (contain untruths) or incomplete (exclude truths).

Godel’s Incompleteness Theorem is breathtakingly simple. First we postulate Axiomatic system A, and state that every true statement, and only true statements, are contained within it. Then we create Statement B, a statement that possesses an interesting quality - self-reference; it talks about itself. And what it says is “B is not an axiom of A.” What has happened here?

If we include Statement B within Axiomatic System A, then A now contains the false statement that B is not an axiom of A, but if we exclude Statement B from Axiomatic System A, then the true statement that B is not an axiom of A lies outside of A, and A therefore does not contain all true statements. B either belongs NEITHER inside NOR outside of A, or it belongs BOTH inside AND outside A (both impossible alternatives, involving the (neither X nor not-X) and the (both X and not-X) logical contradictions, respectively), and the contradiction is not resolveable within Axiomatic System A. The bottom falls out; mathematics is revealed to be fundamentally a Zen koan.